From Serhiy Grabarchuk: Neo Matchstick Snake. You have
a 2x2 square and a pile of matchsticks. (Every matchstick has its length
equal to 1.) The object is to form the longest possible snake exactly
within the 2x2 square (including its border). Matchsticks must not cross
each other, and a final snake must not have self-touches even at a
point. It can be closed in a loop. Angles between the adjacent
matchsticks can be 45, 90, 135 or 180 degrees only. It's easy to find
Snake-12 or Snake-13, but there is even a longer snake. Jeremy Galvagni
and Bob Wainwright both matched my own length
15 solution. Roger Phillips found a beautiful symmetrical
length-15 solution. Jim Lewis Melby, Clinton Weaver, Susan
Hoover, Michael Dufour, and Brian Trial sent length 14 solutions.
If one sticks to multiples of 30 degrees, I couldn't get past
Snake-15. I managed to put a degree-30 Snake-13 entirely inside a
1.95 sided square. Can you find it?

A lot of new small sliding block puzzles
are available at Puzzlebeast.

An extensive site about boardgames is the Games Journal. I was
introduced to it through a 123
part essay on game systems by Ron Hale-Evans. I'm also quite
pleased with my subscription to Abstract Games magazine.
And I'm constantly amazed by the new things invented with Zillions of Games.

Last week: "I recently learned the
importance of the numbers 299792458
and 9192631770.
If you wish, you can read the official paper on units."
Several of you, including myself, were slightly nonplussed about the
international standard of weight being based upon a casting made by George Matthey in 1879.
I was happy to learn of an ongoing project to replace the old standard
with a perfect sphere
of silicon.

I made a bad mistake while drawing up
solutions for Brian Trial's coin problem from 5 February. Not only
did I get the size of the card wrong, but I also calculated the radius
of the dime incorrectly, so the "solution" I found was totally
wrong. (My first clue should have been the matching solutions of
master solvers Clinton Weaver and Bob Wainwright.) My apologies. A
lesson for aspiring math people: always be wary of a solution with
warning flags.

material added 16 March 2003

Pack squares of size 1 to 17 into a
39×46 rectangle. Answer. Solvers. The uncovered area in only 9, which can
be filled with a domino, tromino, and tetromino. Although
suggested to me by Robert Reid, Patrick Hamlyn, and William Rex
Marshall, this puzzle can be traced to online sequence A038666.
It's a very nice problem, I managed to solve it in about 10 minutes.
Only the first 21 cases are known, the excess areas for squares up to
size n are: 1-02-13-14-55-56-87-148-69-1510-2011-712-1713-1714-2015-2516-1617-918-3019-2120-2021-33.

In Thunderball there are n
balls which roll about on the surface of a squared grid. The balls move
in response to a tilting of the surface, and all move at the same time
in the same direction and the same distance, unless one or more of them
are blocked. As the balls move they leave behind them a trail of damaged
squares. Each visited square is so badly damaged that balls are
subsequently blocked from revisiting it; if a tilt would bring a ball on
to a previously visited square, the ball remains stationary. The balls
start in a diagonal row of n,
and the object is to bring them in to a horizontally or vertically
adjacent row of n.

The 2-ball case can be solved by making tilting: N, W, 2S, 2E, 2S, E,
4N, 6W, 6S, 5E, S, E, 2N. This brings the balls in a line within the
confines of an 8x7 rectangle. In fact it is possible to do so within a
5x7 rectangle. (How? Answers and Commentary.)

Can the problem be solved for all n? Well, here is a solution to the
4-ball problem.

I believe that this method of solution can be extended, with suitable
changes of various aspects, to solve the problem for any larger n. This
suggests that the problem is solvable for all n, but the method gives
solutions which are far from minimal. By minimal, I mean a solution in
which the encompassing rectangle has largest side as small as possible.
(So 6x6 would be smaller than 5x7. This definition of minimal penalises
long thin solutions like the 4-ball solution above more than a simple
product of the sides would do.)

This type of problem, where all pieces move simultaneously in the same
direction but squares may not be revisited, permits a great many
variants. Start or target positions could be different, the board could
have edges, or holes, off which the balls are not permitted to fall.
There could be barriers or blocked squares which would help and hinder
progress, or pits which trap balls. There could be anti-gravity balls,
or ones which leave a trail of oil over which others may skid without
stopping. Or the directions could be other than just orthogonal, for
example diagonals could be permitted, or knight moves. I’ve tended to
restrict myself to the pure version of the problem as above.

material added 11 March 2003

I fixed the link to the ISEE3,
so I may as well point out the Shape of Space again as a
separate type of manifold. Stephen Hawking weighed in on this as
well in an episode of the Simpsons:
"Homer, your theory of a donut-shaped universe is intriguing."
According to a recent New
York Times article, Homer may have been right.

material added 9 March 2003

A conference devoted to Mathematical Art
produced many beautiful objects and sculptures, which you can see at the Bridges & Isama
website. I did some experiments in mathematical art myself while looking
at the marvellous Nova
Plexus by Geoff Wyvill. If you don't mind rubber bands, you
can make one of these with 12 pencils and 12 small rubber bands -- it's
quite attractive. If you don't like rubber bands, you can always
use more
pencils.

One good pencil puzzle is to arrange 6
pencils so that they all touch each other. With more work, 7
pencils can all touch each other. With even more work, 8 variously
sharpened pencils can all touch each other. Can you figure out
how? Send answer.

A nice result, from http://www.ktn.freeuk.com/9f.htm
A fiveleaper is a type of generalised knight that makes moves of length
5 units, with coordinates either {0,5} or {3,4}. In Variant Chess, GP
Jelliss made the following observation: “Since the fiveleaper has four
moves at every square of the 8×8 board it follows that in every
closed tour the unused moves are also two at every square, and therefore
form either a tour (is this possible?) or a pseudotour (i.e. a set of
closed circuits). The question of whether such a double tour is
possible was in fact answered in the affirmative by Tom Marlow in a
letter to me of 17 November 1991:

I found an article
about the fantastically complicated journey of the International
Sun-Earth Explorer 3 to be fascinating. I knew about the satellite
years ago, back when I worked at NORAD. Basically, the Lagrange
point between the Earth and Sun is unstable, but there is a stable orbit
around it.

Gary J. Shannon has
posted his investigations in a WireWorld like, logic-gate rich CA at his site.

material added 2 March 2003

Robert Reid sent me his efforts for
packing 16, 17,20, 22, 41, 43, 45, 49, 51, 64, 65, 76, 90, and 94
consectutive squares into a square. In each case, it is impossible
to fit the squares in a smaller square. I redrew two of them. In
general, what are the smallest rectangles than can hold all the squares
up to size n? The squares up to size 42 don't
quite fit into the smallest square.

Joseph DeVincentis: I finally coded up
my idea for a method of searching for the longest words possible in the
DNA code. I started by making a list of all the possible trigrams, by
going through all 4^5 possible 5-base sequences. Then I used some grep
commands to search word lists for words that consisted only of these
trigrams at all positions. The use of trigrams eliminates words that
would erroneously be formed, for instance, by entering 'S' at UCx and
leaving via AGx, if you only used bigrams. The longest word I
found was flytrap which can be written as uUUACGCCu. Also,
entrap, gently, and several less common 6-letter words showed up. Using
the allwords list, I get a few other 7s: Cardiff, prerent, and Sargent,
and one 8, alcargen, which appears in the list which supposedly
represents the Shorter OED, but it is one of those "words" that shows up
nowhere on the web except in word lists. Michael Dufour sent me
CARARARA (an obscure monkey) and Perl code.

I recently found the exact solution for
the Snub
Dodecahedron. I was interested in finding a vectors {1,a,b}
that could be multiplied by the Icosahedral
Group to obtain all of the solids in
that group. Using Mathematica, a horrifying equation boiled down
to Root[x6 + 6 x5 - 7 x4 - 9 x3
- 14 x2 - 7 x - 1 &, 4]. Withh more Mathematica, I obtained
all 169 vectors that produce a convex solid with icosahedral symmetry.
Sixty of these points are for the snub dodecahedron alone. I
haven't yet plumbed all the secrets out of this set of points, but I
rather like it.

Robert Abbott let me know about a Washington
Post article about Binary Arts, Bill Ritchie, and Andrea Gilbert's Clickmazes.com. Serhiy
Grabarchuk, who runs Puzzles.com,
filled me in on how that website is joining in. Andrea's plank
maze will be a key demo at the currently ongoing Toy Fair.

Snopes.com is usually good for accuracy,
but I noticed a scientific blunder on this page about
diamonds. Can you figure out what it is? Mark Thompson
matched my answer: I wouldn't call it a scientific "blunder," but I can
see a way of telling whether a diamond was produce from graphite or from
Uncle Fred. If it's from Uncle Fred, it will have a whole lot more
Carbon 14 in it. Perhaps that could be determined nondestructively with
a geiger counter, though the normal method would require destroying the
gem, in order to verify it.

Brendan Owen made a very nice discovery.
Four corners of a cube can be removed to make a tetrahedron.
If a cube is divided into 4 identical pieces so that each gets an
entire corner, you get pieces he calls cubecorners. If four
cubecorners -- tetracubecorners -- are connected together with
full face connection,one piece that can be made is the original cube.
It turns out there are exactly 27 other shapes that can be made,
and they can fit together to make a cube. After solving all that,
he put them all together in a lovely
applet.

I've spent a lot of time lately trying to
LURN. Left, U-turn, Right, No
turn -- these are the directions a Turmite must
choose from, and I wondered what would happen if a turmite could split
as an action... say Left and Right. In addition, I added the rule
that when two turmites met, they annihilated each other. Some
interesting patterns came out from my initial study. Here is my Mathematica notebook, for those that want to
study them (with some help from Eric
Weisstein's MathWorld packages).
My main interest is finding turmites that will grow for a long
time, then self-annihilate.

I read a paper by George Collins (Lecture
Notes in Comput. Sci., 358). If b and c are two
random integers, then the probabilty that they have no common factor is
six over pi squared. Or P[GCD[b,c]=1] = (6/Pi^2). That's a
well-known nice result. If b and c are two random Gaussian integers, then P[GCD[b,c]=1] =
(6/Pi^2)/Catalan, where Catalan is the Catalan
constant.

Here's one by Robert Reid. Find a three digit number abc such
that abc×bca×cab is a square number. There are two
solutions. As a hint, abc×bca×cab is a sixth
power, in one of the solutions. Answer.

Bathsheba Grossman is one of the
foremost mathematical sculptors at the moment, and I've much enjoyed
looking over the artist's pages. One new project is
particularly fascinating -- large sculptors from laser cut materials.
You can get the Sea Star project
for just $22. The techniques Bathsheba uses are also discussed --
most design work is done with Rhinoceros.
That program is well worth your valuable time for a look --
although it costs $900, the only limit on the demo version is that only
25 saves can be made.

Brian Trial: Take a 2 1/2 inch by 4 1/4
inch card and cover it with as many U.S. pennies, nickles, dimes, and
quarters as you can to get the highest dollar amount. Coins must lie
flat, must not overlap or stack onto other coins in any way, and must
lie entirely on the card. For reference, a U.S. quarter has a diameter
of 0.955 inches, a nickle has a diameter of 0.835 inches, a penny has a
diameter of 0.750 inches, and a dime has a diameter of 0.705
inches. Answers. (No-one,
including the problem creator, got the correct
answer.) Now, that's an awfully US-centric puzzle, so here's a more
international version -- using the coins of your country, what is the
most money in coins you can place on a 7cm by 11cm card, using the same
rules?

Jean-Charles Meyrignac: After more
than 3 months of computation, I just finished the computation to the c1
solitaire problem on the french board. In the book Ins and Outs of Peg Solitaire from
D.J. Beasley, it is mentioned that it is possible in 21 moves. In fact,
I discovered that it can be done in 20 moves, and the solutions are very
rare (only 280). More Info.

Geometry In Action
Java Gallery, by Clark Kimberling, is well worth a look. I
think of all the courses I took in high school, Geometry was the one I
found most useful, both in terms of proof technique and the general
usefulness of geometric construction.

Bob Lukes has created the Lonely Unit
Cube puzzle out of wood. Bob says that, after all the work he went
through cutting and painting the cubes, that if anyone wants a copy, the
answer is "no". It is possible, though, to get 54 8mm and 24 12mm
dice, in which case you'd only need to make 1 4mm cube, 2 20mm cubes,
and a 44mm box. Any volunteers to make that? I'd like to put
together about 20 sets for general purchase.

I have updated the Neglected Gaussian page with many solutions
by Fred Helenius and W. Edwin Clark.

material added 26 January 2003

If the edges wrap, can a set of double-6
dominoes be placed in a 8x7 rectangle so that all numbers are in
connected groups? The below is one of my efforts -- the threes are all
connected if you consider the edges as wrapping, same with the blanks
and sixes. However, the ones and twos are both in disconnected
groups. Can everything be connected? It turns out the answer is yes. The solution below, by Jason
Woolever, is for the harder problem of connecting everything without the
use of doubles. Other answers and solvers.
See also my 2
September 2001 update for a related set of solutions by Roger
Phillips. The domino-connection number for the torus is thus 7.
Suppose we go to 1x1x2 blocks as dominoes. What is the domino
connection number for a 3-D block? If the faces of a block wrap,
it is called a 3-torus, or a 3-manifold. You can learn more about
these at The Shape of Space.
Without doubles, what is the domino connection number of the
3-torus? With doubles? With 2 of each double? In 3D
space, as the number of doubles increases, the domino connection number
would go to infinity (why?), but I don't know if anyone has looked at
how fast. Send Answers.

The cubicular goodness of the 2003 MIT
Mystery Hunt (acme-corp.com)
can be seen as an offshoot of the main MIT Mystery page. All the
puzzles are there for your perusal.

Tom Marlow notes two interesting squares:
4253907186^2 = 1809572634 7102438596, 5296031874^2 = 2804795361
0423951876. Daniel Scher's Geometry in Motion has
moved. Bob
Kraus has put Soccolot on the ZOG site (interesting
game). Martin
Watson has add lots of great stuff. Ivar's Peterson has had a
number of great columns lately, such as his Dearth of Primes
write-up (I had no knowledge of this).

An even more interesting game is Amazons,
and there is a very nice analysis of it at the More
Games of No Chance page. The full book is available online, but
I've seen so far is good enough to prompt me to buy
it.

Cihan Altay has started PQRST 4. There are
many clever ideas here, I especially like Puzzle #8. Answers must
be submitted by 18 January.

Robert Reid found a mistake in one of the
puzzles on my old Solution page. "The first puzzle this week is by Scott
Purdy. The thick path travels from A to B, visiting every
dot. Can you remove 7 of the thin lines so that this is the only path from A to B that visits each
dot? In more mathematical terms, for K(n), what is the minimal
number of edges that needs to be removed for a unique Hamiltonean path
between two given points? The case for K(8) is unsolved (as far as
I know). Partial
solution to the general case by
Scott Purdy and Erich Friedman." Okay, seems okay -- but
Robert Reid found a solution that removes only 6 lines ... and solved
the K(8) caseby removing only 9 lines. Can you find Reid's six-line
solution? Only Jim Boyce matched Reid's answer.

1 2 3 4 5 6 7 8 = 2003. Add each of
+, -, ×, ÷ exactly once to
make the equation true. Answer.
This is by Yoshio Mimura.
I found his page while looking to see if anyone else had noticed the
octal square 177771777177771.

So ...
177771777177771. Twenty years ago, Nob Yoshigahara noticed that
81619 × 81619
= 6661661161. Are there larger square numbers using only two
digits? No-one knows. The
Mathematician Secret Room has more data for the 3-digit square problem.
I wondered if I could find new complex squares with that property
(I couldn't), or two digit squares in other bases (easy). Nick
Baxter looked at rational squares of two digits, and found some
interesting solutions. There is enough here to figure out the
significance of 34343443434344.

Robert Henderson has found a remarkable
Latin pentacube solution. With the following division of a 5x5x5
cube into 25 different pentacubes, find a a way to color the cube in 5
colors so that every row, column, stack, and pentacube is comprised of
all 5 colors. Here is the answer.

Some prime numbers remain primes when reversed. Are there an
infinite number of them? John Gowland sent me the following cross-number
puzzle called Reversed Squares, which is based on reversible primes.
Here are some clues, a solving strategy, and the answer.

material added 5 January 2003

Those who have Zillions of Games can try
out Bob Kraus's
very nice Extraction puzzles, now at the Zillions site.
Eventually, some brave soul is going to need to look at all of the
Zillions files that are available, and summarize what is best.
That will be hard, because most of the files are quite good, and
time-absorbing.

Erich Friedman:
Sometimes a day is the sum of the digits of the horizontally and
vertically adjacent days on that month's calendar. For example, this
happened on december 14, 22, 24, and 26 of 2002. There is only one day
in 2003 on which this will happen but be the ONLY time that month it
will happen. When
is it?

Here are some small fractions that make a
nice approximation to a familiar number. 22/17 + 37/47 + 88/83 !=
Pi. Can anyone find a better approximation with small fractions?
I've noticed the greedy algorithm doesn't work very well at
finding better solutions.

Some things, I've been keeping around.
For example -- here's a little puzzle -- I can't remember if I
made it, and if I did make it, I can't remember the answer, and I
haven't solve it. So it wouldn't be all that fair to present it,
would it? Well, here it is, anyways. And here is a whole page of other material I
never quite figured out how to present, so here it is, all in one big
batch. A similarly disorganized page is my g4g5
writeup. Comments are
welcome. Answer to 4-divide
(Solutions sent by Remmert Borst, Cihan Altay, Paul Cooper, Jonathan
Welton, Jon k McLean, Franz Pichler, Clinton Weaver, Agaeus, Matt Elder,
Joseph DeVincentis, Kirk Bresniker, Jeremy Galvagni, and Juha
Hyvönen) If you like division puzzles, I don't think anyone has
solved all of Mike Reid's puzzles.

The various integer-sided blocks with
sides 0<a<b<c<7 will fit into a 7x7x15 block, as shown by Erich Friedman.

Warning: Big File! Edward Brisse
compiled all of the Triangle Centers into one big text file. I
won't give a direct link, but you can find it at EdwardBrisse.txt, or
EdwardBrisse.zip, on this site.

material added 31 December 2002

While playing around with Pick's Theorem,
I came up with a deceptively tricky little problem. Divide a 5x5 square
into 5 regions which have identical perimeters but differing
areas. All lines must be straight, and must connect grid vertices. Answers. Another by Livio
Zucca. I used a trick in my solution (1) -- I was surprised when
Joseph Devincentis sent me different solution (2): "This was a very nice
puzzle, and a wonderful demonstration of the usefulness of Pick's
Theorem. With the theorem I could quickly see that I needed to somehow
split up the regions using lines that crossed only 6 of the internal
points, with the five regions containing 0,1,2,3,4 of the other internal
points. It still took me a while to find the correct perimeter and
arrangement of lines to make it work." After that, Daniel Scher
and Martin Bernstein sent solutions (3) and (4). Taus
Brock-Nannestad sent a 5th answer, and William sent 20 more (using the
3-4-5 triangle trick in a normal grid).

Theo Gray and I managed to collect
all 90 stable elements. We have started testing some of the
samples. One of the most bizarre -- a weird
rock I found when I was six years old has turned out to be 38%
titanium. On the other hand, a "Titanium" tennis racket wound up having
no titanium whatsoever.

Stephen Wolfram has made some of the Historical Notes
from A New Kind of Science
available. I like what he does with WireWorld.

NetLogo 1.2 has been
freely released at the Center for Connected Learning and Computer-Based
Modeling, at Northwestern University. The Logo language is
frequently known as the language for "programmable turtles." A
turtle with the instruction set {move 1, turn 90 degrees} would make a
square. There is much, much more, and this releases is filled with
lots of excellently documented programs in art, biology, chemisty,
physics, computer science, earth science, mathematics, social science,
and more. No programming knowledge is necessary, you can just start up
the models and what how they work. With slight programming, the models
are easily modified. It's a wonderful package for learning.

Jean-Charles Meyrignac found an old puzzle
that involved the 18 ways to three-color a tromino. How many can be
placed in a 7x7 square so that only like color touch? It seems 13
is the answer. In an 8x8 square, he can place all but 1 piece, and isn't
sure if all 18 is possible. He wrote a program to find how many of the
40 4-colored trominoes could be placed in an 11x11 square, and he found
two solutions with 33 pieces placed. He does not know if these can
be improved. It's a nice task to split the grids into the
different trominoes.

I rather like Pascal's Triangle at The Sound of Mathematics.
More algorithmic music is at Tune Toy.
Somewhere on my site I need to update details of Don Wood's 20 questions
-- a fixed version is at his site.
Listening to various carols recently, I noticed a six letter word
with 18 syllables. Kevin Wald noticed the same word is sung with 21
syllables in "Ding Dong Merrily Along." There is a 36-syllable
five-letter word in "Poor Wandering One" from Gilbert and Sullivan's Pirates of Penzance. Beating that, in Handel's "For
Unto Us a Child Is Born," the word "born" is sung with 57 syllables.
Gordon Bower: "The longest one of these that comes to mind is from
Mozart's Magic Flute, about two-thirds of the way through the Queen of
the Night's aria "Der Hoelle Rache kocht in meinem Herzen": "Bande" gets
86 notes in the queen's part (85 for Ban- and 1 for -de). Admittedly
some of these are slurred together, so I suppose it is only 70-ish
syllables."

The CRC
Concise Encyclopedia of Mathematics, Second Edition by Eric
Weisstein is now available. If you use the link I provide, Eric will get
a percentage of every sale. It gives a good glimpse at the scope
of his MathWorld site.
Hopefully, in a future edition, CRC will give Eric standard royalties
for his work, and full control over how it is printed. That would
allow for a much better book, more up-to-date, with better pictures.
It's very sad that CRC has done so much to give Eric a hard time about
the material he created.

In contrast, over the past few weeks I've been wending my way through Mummy Maze Deluxe. The
game has an interesting history. Originally, Robert Abbott designed the Theseus
and Minotaur maze for Mad
Mazes, which was later turned into a Java applet by Toby
Nelson. Popcap inadvertantly copied the idea for their game Mummy Maze.
Soon after realizing their error, the Popcap company apologized to
Robert, paid him, and now are giving him credit.
Legally, they didn't have to do any of that. Some of the mazes in
MMD are quite tricky, if you get stuck, you can use the bardavid solver. Some reviews.

Fred W. Helenius noticed that 2542645806624 is
100101000000000001100000000000011000100000 in binary, and
100000002000100200000002000 in ternary. At most 4/21st of the
digits in the ternary/binary representations are non-zero. Is there a
positive integer with a lower percentage of non-zero digits in both the
binary and ternary forms? My best finds were 1208614932, 2453670144,
17448310278, and 22083026976. Send
Answer. Update: Robert Harley search up to 2^64, and
found 451521135633235968, which gives
11001000100001000000001010010100010000000000000000000000000 and
10000020000020200000020000000001000000.

Martin Watson rediscovered a nice puzzle -- pack 11 F pentominoes into
an 8x8 square, or a 4x4x4 cube. Both solutions are unique. The puzzle is
actually sold commerially, at Polzeath, Cornwall. Patrick Hamlyn
found the puzzle to be too easy for computer solving, so he offers a
counter challenge: Fit 20 solid Z-pentominos plus five other pentominoes
into a 5x5x5 cube. Send Answer.

Martin Gardner celebrates math puzzles and
Mathematical Recreations. This site aims to do the same. If you've
made a good, new math puzzle, send
it to ed@mathpuzzle.com. My mail address is Ed Pegg Jr, 1607
Park Haven, Champaign, IL 61820.You can join my recreational
mathematics email list by sending email to majordomo@wolfram.com(with subscribe as the subject and subscribe
mathpuzzle as the body). Other math mailing lists can be
found here.

All material on this
site is copyright 1998-2002 by Ed Pegg Jr. Copyrights of
submitted materials stays with contributor and is used with permission.
visitors since I started keep track. Yes, over one million.